Field of Invention
The present invention relates to a technical field of measurement, and more particularly to an in situ spacing-correcting heat pulse sensor based on both inline deflections and non-inline deflections for measuring sap flow or water flux and a measuring method thereof.
Description of Related Works
1. Theoretical Background of Soil Water Flux Measurement
A device for measuring soil water flux with the heat pulse method comprises three probes (with higher thermal conductivity) in parallel, wherein a heating probe is in the middle, and the other two are temperature probes with thermometric elements, which are arranged on both sides of the heating probe. The temperature probes include an upstream temperature probe and a downstream temperature probe as shown in FIG. 1. The soil water flux can be estimated by maximum difference of the temperatures recorded by the upstream and the downstream temperature probes.
For water flow with constant velocity in a homogenous and infinite medium, the heat conduction equation is:
                                          ∂            T                                ∂            t                          =                              α            ⁡                          (                                                                                          ∂                      2                                        ⁢                    T                                                        ∂                                          x                      2                                                                      +                                                                            ∂                      2                                        ⁢                    T                                                        ∂                                          y                      2                                                                                  )                                -                      V            ⁢                                          ∂                T                                            ∂                x                                                                        (        1        )            wherein, T is temperature (° C.), t is time (s), α represents thermal diffusivity (m s−2), and x and y represent spatial coordinates.V=θwVwCw/C=JCw/C  (2)wherein, θw represents water content, Vw represents water flow rate (m s−1), Cw represents volume heat capacity of water (J m−3° C.−1) and C represents volume heat capacity of soil (J m−3° C.−1). The heat conduction equation illustrates the relationship between the thermal diffusivity and V. With a known thermal diffusivity, V is able to be calculated with measured data. Then the water flux J is able to be calculated according to the equation 2.
For a homogeneous medium, when heated with an infinite linear source for a heating time of t0, temperature change of the temperature probe satisfies:
                              T          ⁡                      (                          x              ,              y              ,              t                        )                          =                  {                                                                                                                q                                              4                        ⁢                        πλ                                                              ⁢                                                                  ∫                        0                        t                                            ⁢                                                                        s                                                      -                            1                                                                          ⁢                                                  exp                          ⁡                                                      [                                                          -                                                                                                                                                                          (                                                                              x                                        -                                        Vs                                                                            )                                                                        2                                                                    +                                                                      y                                    2                                                                                                                                    4                                  ⁢                                  α                                  ⁢                                                                                                                                          ⁢                                  s                                                                                                                      ]                                                                          ⁢                                                  ⅆ                          s                                                ⁢                                                                                                  ⁢                        0                                                                              <                  t                  <                                      t                    0                                                                                                                                                                  q                                              4                        ⁢                        πλ                                                              ⁢                                                                  ∫                                                  t                          -                                                      t                            0                                                                          t                                            ⁢                                                                        s                                                      -                            1                                                                          ⁢                                                  exp                          ⁡                                                      [                                                          -                                                                                                                                                                          (                                                                              x                                        -                                        Vs                                                                            )                                                                        2                                                                    +                                                                      y                                    2                                                                                                                                    4                                  ⁢                                  α                                  ⁢                                                                                                                                          ⁢                                  s                                                                                                                      ]                                                                          ⁢                                                  ⅆ                          s                                                ⁢                                                                                                  ⁢                        t                                                                              >                                      t                    0                                                                                                          (        3        )            wherein, s represents dummy variable for evaluating integral, q is heat released by a unit length of the linear heat source within a unit time (J m−1 s−1).
Conventional methods for calculating soil water flux:
a) Maximum Dimensionless Temperature Difference Method, i.e. MDTD Method
The maximum dimensionless temperature difference (DTD) is defined as:
                    DTD        =                              4            ⁢                          πλ              ⁡                              (                                                      T                    d                                    -                                      T                    u                                                  )                                              q                                    (        4        )            wherein Td is the temperature increment of the downstream temperature probe, Tu is the temperature increment of the upstream temperature probe. When reaching the maximum value, the DTD is marked as MDTD.
                    MDTD        =                              ∫                                          t                m                            -                              t                0                                                    t              m                                ⁢                                    s                              -                1                                      ⁢                          {                                                exp                  ⁡                                      [                                          -                                                                                                    (                                                                                          x                                d                                                            -                              Vs                                                        )                                                    2                                                                          4                          ⁢                          α                          ⁢                                                                                                          ⁢                          s                                                                                      ]                                                  -                                  exp                  ⁡                                      [                                          -                                                                                                    (                                                                                          x                                u                                                            +                              Vs                                                        )                                                    2                                                                          4                          ⁢                          α                          ⁢                                                                                                          ⁢                          s                                                                                      ]                                                              }                        ⁢                          ⅆ              s                                                          (        5        )            wherein tm is the time when the DTD reaches the maximum value, t0 is the heating time, xu is the spacing between the upstream temperature probe and the heating probe (actually, the measured xu is the spacing between the temperature element in the upstream temperature probe and the heating probe), and xu is the spacing between the downstream temperature probe and the heat source (also as the upstream, the measured xu is the spacing between the temperature element in the downstream temperature probe and the heat source). After obtaining the MDTD value by measurement, V is able to be calculated according to the calculated thermal diffusivity, and then the water flux is obtained according to the equation 2.
b) Heat Ratio Method
It is introduced that the water flux can be calculated according to the ratio of the temperature increments of the two temperature probes. Accordingly, a calculation equation for water flux J is:
                              J          =                                                                      2                  ⁢                  λ                                                                      (                                                                  x                        d                                            +                                              x                        u                                                              )                                    ⁢                                      C                    w                                                              ⁢                              ln                ⁡                                  (                                                            T                      d                                                              T                      u                                                        )                                                      =                                          C                ⁡                                  (                                                            x                      d                                        -                                          x                      u                                                        )                                                            2                ⁢                                                      C                    w                                    ⁡                                      (                                          t                      -                                              (                                                                              t                            0                                                    ⁢                                                      /                                                    ⁢                          2                                                )                                                              )                                                                                      ;                  t          >                                    t              0                        ⁢                          /                        ⁢            2                                              (        6        )            
c) Maximum Temperature Time Method
Supposing that the time when the measured temperature of upstream temperature probe reaches the maximum temperature is tu, and the time when the measured temperature of temperature probe reaches the maximum temperature is t*, then the relationship is:
                              V          2                =                                                            4                ⁢                α                                            t                0                                      ⁢                          ln              ⁡                              (                                                                            t                      *                                        -                                          t                      0                                                                            t                    *                                                  )                                              +                                    x              *              2                                                      (                                                      t                    *                                    -                                      t                    0                                                  )                            ⁢                              t                *                                                                        (        7        )            where t* can be td or tu respectively and correspondingly, x* can be xd or xu respectively.
V is obtained according to the above equations. Combined with the thermal diffusivity and the equation 2, the water flux J can be obtained.
2. Theoretical Background of Sap Flow Measurement
Basic principles of a heat pulse method for measuring sap flow in trees are similar to those for measuring soil water flux. Probes thereof are basically the same as the probes for measuring soil water flux, but differ in the probe spacings and the probe lengths. Initial spacings of the probes for measuring soil water flux, xu and xd, are approximately the same, wherein the initial spacings are about 6 mm. For the sap flow measurement, great differences exist between different methods. Several popular methods for measuring sap flow density (SFD) (m3 m−2 s−1) are mainly introduced here.
For an ideal instantaneous line source, temperature changes thereof satisfy:
                              Δ          ⁢                                          ⁢          T                =                              q                          4              ⁢              πλ              ⁢                                                          ⁢              t                                ⁢                      exp            ⁡                          (                                                -                                                            ρ                      ⁢                                                                                          ⁢                      c                                                              4                      ⁢                      λ                      ⁢                                                                                          ⁢                      t                                                                      ⁢                                  (                                                                                    (                                                  x                          -                                                                                    V                              h                                                        ⁢                            t                                                                          )                                            2                                        +                                          y                      2                                                        )                                            )                                                          (        8        )            wherein the coordinate of the heat source is (0, 0), ΔT is the temperature change at the point (x, y) after applying a heat pulse with a strength of q (m s−1), Vh is heat pulse velocity (m s−1). Unlike the soil, the tree has different axial and tangential heat conductivities. Therefore:
                              Δ          ⁢                                          ⁢          T                =                  {                                                                                                                q                                              4                        ⁢                        π                        ⁢                                                                                                            λ                              ax                                                        ⁢                                                          λ                              tg                                                                                                                                            ⁢                                                                  ∫                        0                        t                                            ⁢                                                                        1                          t                                                ⁢                                                  exp                          ⁡                                                      [                                                                                          -                                                                                                      ρ                                    ⁢                                                                                                                                                  ⁢                                    c                                                                                                        4                                    ⁢                                    t                                                                                                                              ⁢                                                              (                                                                                                                                                                                    (                                                                                  x                                          -                                                                                                                                    V                                              h                                                                                        ⁢                                            t                                                                                                                          )                                                                            2                                                                                                              λ                                      ax                                                                                                        +                                                                                                            y                                      2                                                                                                              λ                                      tg                                                                                                                                      )                                                                                      ]                                                                          ⁢                                                  ⅆ                          t                                                ⁢                                                                                                  ⁢                        0                                                                              <                  t                  <                                      t                    0                                                                                                                                                                  q                                              4                        ⁢                        π                        ⁢                                                                                                            λ                              ax                                                        ⁢                                                          λ                              tg                                                                                                                                            ⁢                                                                  ∫                                                  t                          -                                                      t                            0                                                                          t                                            ⁢                                                                        1                          t                                                ⁢                                                  exp                          ⁡                                                      [                                                                                          -                                                                                                      ρ                                    ⁢                                                                                                                                                  ⁢                                    c                                                                                                        4                                    ⁢                                    t                                                                                                                              ⁢                                                              (                                                                                                                                                                                    (                                                                                  x                                          -                                                                                                                                    V                                              h                                                                                        ⁢                                            t                                                                                                                          )                                                                            2                                                                                                              λ                                      ax                                                                                                        +                                                                                                            y                                      2                                                                                                              λ                                      tg                                                                                                                                      )                                                                                      ]                                                                          ⁢                                                  ⅆ                          t                                                ⁢                                                                                                  ⁢                        t                                                                              >                                      t                    0                                                                                                          (        9        )            wherein λax is the axial heat conductivity of the wood, λtg is the tangential heat conductivity of the wood.
The SFD can be calculated according to the following equation:
                    SFD        =                                            ρ              d                                      ρ              s                                ⁢                      (                          MC              +                                                c                  dw                                                  c                  s                                                      )                    ⁢                      V            h                                              (        10        )            wherein, MC is the water content of the wood (weight of water over weight of dry wood), cdw is the specific heat of the wood (J kg−1 K−1), ρd is the density of the dry wood (kg m−3), ρs is the density of the sap (kg m−3), which is usually considered as the density of water, and cs is the specific heat of the sap flow (J kg−1 K−1), which is usually considered as the specific heat of water.
a) Compensation Heat Pulse Method, i.e. CHP Method
                              V          h                =                                            x              d                        -                          x              u                                            2            ⁢                          t              c                                                          (        11        )            wherein, tc is a time when the temperature measured by upstream and the downstream temperature probes reach the same value. According to the method, differ from the measurement of the soil water flux, xd and xu are different initially.
b) Tmax Method:
In the Tmax method, only one temperature probe, the downstream temperature probe, is utilized. The temperature detected by the temperature probe reaches the maximum value at time td:
                              V          h                =                                                                              4                  ⁢                                      α                    ax                                                                    t                  0                                            ⁢                              ln                ⁡                                  (                                      1                    -                                                                  t                        0                                                                    t                        d                                                                              )                                                      +                                          x                d                2                                                              t                  d                                ⁡                                  (                                                            t                      d                                        -                                          t                      0                                                        )                                                                                        (        12        )                                          α          ax                =                                            x              d              2                                      4              ⁢                              t                d                                              ⁢                                                                      t                  0                                                  (                                                            t                      d                                        -                                          t                      0                                                        )                                            ⁡                              [                                  ln                  ⁡                                      (                                                                  t                        d                                                                                              t                          d                                                -                                                  t                          0                                                                                      )                                                  ]                                                    -              1                                                          (        13        )            wherein αax represents axial thermal diffusivity of wood when the sap flow is 0.
c) Heat Ratio Method, i.e. HR Method
Calculation of the heat ratio method is based on that xd and xu are both equal. As a fact, xd and xu are easy to change. Therefore, the improved equation is:
                              V          h                =                                            4              ⁢                              α                ax                            ⁢              t              ⁢                                                          ⁢                              ln                ⁡                                  (                                      Δ                    ⁢                                                                                  ⁢                                          T                      d                                        ⁢                                          /                                        ⁢                    Δ                    ⁢                                                                                  ⁢                                          T                      u                                                        )                                                      -                          x              d              2                        +                          x              u              2                                            2            ⁢                          t              ⁡                              (                                                      x                    u                                    -                                      x                    d                                                  )                                                                        (        14        )            wherein t is the time when the ratio of the temperature change of downstream and upstream temperature probe (ΔTd/ΔTu) becomes a nearly invariable value (the average temperature ratio of measurements between 60 and 100 s can satisfy that). Some people use the median measurement time (80 s) or average solutions calculated for each t value used in the measurement series.
d) “Sap Flow+” Method
According to the method, for the upstream and downstream temperature probes, there is a horizontal temperature probe instead of the upstream temperature probe in some cases, wherein the spacing between the horizontal temperature probe and the heating probe is y. According to the equation 9, desired parameters can be obtained.
There are still many other methods for measuring sap flow in trees, without further introduction herein.
3. Calculation of Thermal Diffusivity α
Only when the sap or water flow rate is approximately 0 are the thermal diffusivity and the specific heat able to be accurately measured. Specific heat measurement is an approximate solution based on infinite linear heat source (ILS). When the linear heat source is infinite and heats continuously from 0 to t0, given that the space between the heating probe and the temperature probe is r (it can be xu or xd), the temperature increment can be calculated according to the following equation:
                              Δ          ⁢                                          ⁢                      T            ⁡                          (                              r                ,                t                            )                                      =                  {                                                                                                                -                                              q                                                  4                          ⁢                          παρ                          ⁢                                                                                                          ⁢                          c                                                                                      ⁢                    E                    ⁢                                                                                  ⁢                                          i                      ⁡                                              (                                                                              -                                                          r                              2                                                                                                            4                            ⁢                            α                            ⁢                                                                                                                  ⁢                            t                                                                          )                                                              ⁢                                                                                  ⁢                    t                                    <                                      t                    0                                                                                                                                                                  -                                              q                                                  4                          ⁢                          παρ                          ⁢                                                                                                          ⁢                          c                                                                                      ⁢                                          {                                                                        E                          ⁢                                                                                                          ⁢                                                      i                            ⁡                                                          [                                                                                                -                                                                      r                                    2                                                                                                                                    4                                  ⁢                                  α                                  ⁢                                                                                                                                          ⁢                                                                      (                                                                          t                                      -                                                                              t                                        0                                                                                                              )                                                                                                                              ]                                                                                                      -                                                  E                          ⁢                                                                                                          ⁢                                                      i                            ⁡                                                          (                                                                                                -                                                                      r                                    2                                                                                                                                    4                                  ⁢                                  α                                  ⁢                                                                                                                                          ⁢                                  t                                                                                            )                                                                                                                          ⁢                                                                                          }                                        ⁢                                                                                  ⁢                    t                                    >                                      t                    0                                                                                                          (        15        )            wherein ΔT is the temperature change value (° C.), Ei(−x) is an exponential integral. According to the above equation, a time of tm is needed for the temperature to reach a maximum value, then α can be calculated according to equation 16.
                    α        =                                            r              2                        4                    ⁢                      {                                                            1                  ⁢                                      /                                    ⁢                                      (                                                                  t                        m                                            -                                              t                        0                                                              )                                                  -                                  1                  ⁢                                      /                                    ⁢                                      t                    m                                                                              ln                ⁡                                  [                                                            t                      m                                        ⁢                                          /                                        ⁢                                          (                                                                        t                          m                                                -                                                  t                          0                                                                    )                                                        ]                                                      }                                              (        16        )            wherein, tm is the corresponding time when the maximum temperature increment ΔTm occurs.
Referring to the above introduction of the conventional technology, it is known that no matter which method is utilized for calculating sap or water flux, xu and xd are always the key factors. However, in practice, temperature probes of both sap flow measurement and soil water flux measurement are liable to deform under factors such as external force. For example, during the measurement of sap flow in trees, if pretreatments such as drilling holes are applied, the spacings between the temperature probes (actually the temperature elements) and heating probe is also liable to deviate from the original spacing because of technique defects. Besides, factors such as growth of the tree also matters. The above factors will finally lead to huge deviation of the result of sap flow measurement. During the measurement of soil water flux, due to impurities, such as stones and roots in soil, alternating freezing and thawing, distortion during installation, etc., the temperature probe or the heating probe may be deflect. All of the above factors can change xu and xd, and finally lead to a huge difference between a calculated value and an actual measured value of the sap flow or soil water flux using heat pulse method.